Invariants
| Base field: | $\F_{23}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 7 x + 77 x^{2} - 314 x^{3} + 1771 x^{4} - 3703 x^{5} + 12167 x^{6}$ |
| Frobenius angles: | $\pm0.319926650127$, $\pm0.416585689026$, $\pm0.520795862164$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.41578490624.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9992$ | $180255680$ | $1850971037600$ | $21846195291008000$ | $266434961211355131352$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $635$ | $12500$ | $278967$ | $6431507$ | $148034780$ | $3404801389$ | $78310885007$ | $1801154750300$ | $41426520702675$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 17 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+19 x^7+x^6+16 x^5+12 x^4+18 x^3+11 x^2+5 x$
- $y^2=22 x^8+15 x^7+20 x^6+14 x^5+15 x^4+21 x^3+22 x^2+5 x$
- $y^2=x^8+13 x^7+21 x^6+20 x^5+20 x^4+11 x^3+5 x^2+10 x$
- $y^2=x^8+9 x^7+5 x^6+17 x^5+3 x^4+17 x^3+12 x^2+18 x$
- $y^2=22 x^8+5 x^6+3 x^4+15 x^3+20 x^2+7 x+16$
- $y^2=22 x^8+20 x^7+19 x^6+15 x^5+2 x^4+20 x^3+9 x^2+x+2$
- $y^2=x^8+7 x^7+14 x^6+6 x^5+3 x^4+5 x^3+10 x+20$
- $y^2=x^8+15 x^7+13 x^6+12 x^5+4 x^4+12 x^3+14 x^2+5 x+3$
- $y^2=22 x^7+13 x^6+5 x^5+4 x^4+19 x^3+17 x^2+x+20$
- $y^2=x^8+9 x^7+11 x^6+20 x^5+3 x^4+16 x^3+21 x^2+x+22$
- $y^2=22 x^8+9 x^7+2 x^6+8 x^5+5 x^4+15 x^3+19 x^2+21 x+12$
- $y^2=22 x^8+12 x^7+8 x^6+3 x^5+9 x^4+21 x^3+4 x^2+11 x+14$
- $y^2=x^8+13 x^7+21 x^6+15 x^5+14 x^4+17 x^3+2 x^2+7 x+13$
- $y^2=22 x^8+8 x^7+16 x^6+17 x^5+8 x^4+18 x^3+6 x^2+21 x+3$
- $y^2=22 x^8+2 x^7+17 x^6+5 x^5+12 x^4+2 x^3+3 x+11$
- $y^2=22 x^8+15 x^7+13 x^6+5 x^5+13 x^4+5 x^3+4 x^2+6 x+5$
- $y^2=22 x^8+7 x^7+11 x^6+15 x^5+15 x^4+6 x^3+8 x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$| The endomorphism algebra of this simple isogeny class is 6.0.41578490624.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.23.h_cz_mc | $2$ | (not in LMFDB) |